A038334 → basis A008277

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:             400,              384,                64;
  1:            8000,            13952,              5376,                512;
  2:          160000,           454656,            305152,              65536,               4096;
  3:         3200000,         14010368,          14622720,            5324800,             737280,             32768;
  4:        64000000,        417890304,         638500864,          351272960,           80936960,           7864320,            262144;
  5:      1280000000,      12212928512,       26329153536,        20564017152,         7018905600,        1119354880,          80740352,          2097152;
  6:     25600000000,     352201998336,     1045552955392,      1115449982976,       529495228416,      123312537600,       14445182976,        805306368,        16777216;
  7:    512000000000,   10066455953408,    40457522380800,     57444222894080,     36457351741440,    11634714083328,     1968772743168,     176764747776,      7851737088,      134217728;
  8:  10240000000000,  285956766695424,  1537002453336064,   2851205986385920,   2355336073707520,   989741658931200,   226954259202048,   29184302776320,   2073663897600,    75161927680,   1073741824;
  9: 204800000000000, 8088709467471872, 57619742453661696, 137749083027668992, 145287123723878400, 78227188125532160, 23350822897188864, 4033641084616704, 407662189608960, 23504208527360, 708669603840, 8589934592;

The row polynomials satisfy P(n,x) = (20+8*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 400 + 384*x + 64*x^2.

P[0, x_] := 400 + 384*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 8*x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 400 + 384*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 8*x)*v[i - 1] + 8*x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-cdf5d4e7a8154a31
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-cdf5d4e7a8154a31
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-cdf5d4e7a8154a31

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A008277 e-cdf5d4e7a8154a31

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-f4f0736bd14a35db root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers periodic_product e-b445a5ff4f115ce0 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A038220 → basis A008277

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741